Problem: Below is the graph of $y = a \sin (bx + c)$ for some positive constants $a,$ $b,$ and $c.$  Find the smallest possible value of $c.$

[asy]import TrigMacros;

size(300);

real f(real x)
{
	return 2*sin(4*x + pi/2);
}

draw(graph(f,-pi,pi,n=700,join=operator ..),red);
trig_axes(-pi,pi,-3,3,pi/2,1);
layer();
rm_trig_labels(-2,2, 2);

label("$1$", (0,1), E);
label("$2$", (0,2), E);
label("$-1$", (0,-1), E);
label("$-2$", (0,-2), E);
[/asy]
Answer: We see that the graph reaches a maximum at $x = 0.$  The graph of $y = \sin x$ first reaches a maximum at $x = \frac{\pi}{2}$ for positive values of $x,$ so $c = \boxed{\frac{\pi}{2}}.$